The 3-component mixture of power distributions under Bayesian paradigm with application of life span of fatigue fracture

Mixture distributions are naturally extra attractive to model the heterogeneous environment of processes in reliability analysis than simple probability models. This focus of the study is to develop and Bayesian inference on the 3-component mixture of power distributions. Under symmetric and asymmetric loss functions, the Bayes estimators and posterior risk using priors are derived. The presentation of Bayes estimators for various sample sizes and test termination time (a fact of time after that test is terminated) is examined in this article. To assess the performance of Bayes estimators in terms of posterior risks, a Monte Carlo simulation along with real data study is presented.


Posterior distribution assuming uniform prior (UP)
If no prior or additional prior knowledge is given, the use of UP and JP (Jeffreys' prior) as NIPs are recommended in Bayesian estimation.We assume the uniform (0, ∞) for component parameter m (m = 1, 2, 3) and the uniform (0, 1) for the proportion parameter p s (s = 1, 2) .The joint prior distribution is π 1 ( ) ∝ 1 .Thus, the joint posterior distribution given censored data y is: w h e r e A 11 = 1 + u ,
(5) L ; y ∝  After simplification, the marginal posterior distributions are derived as: Posterior distribution assuming Jeffreys' prior (JP) Jeffreys 17,18 suggested a formula for finding the JP as: , where m is Fisher's information.Here, we take prior distributions of p s (s = 1, 2) are uniform (0, 1) .So, the joint posterior distribution given censored data y using π 2 ( ) ∝ 1 1 2 3 as joint prior distribution is: where The marginal posterior distributions are derived as: (9) As an IP, we assume Gamma(a m , b m ) for parameter m and Bivariate Beta(a, b, c) for the proportion parameter p s .The joint prior distribution is: So, the joint posterior distribution given censored data y is: w h e r e A 13 = a 1 + u 1 , The marginal posterior distributions are derived as:

Bayesian estimation using loss functions
Here, we derived the Bayes estimators (BEs) and their respective posterior risks (PRs) using Squared error loss function (SELF) and quadratic loss function (QLF) as symmetric loss functions, whereas, DeGroot loss function (DLF) and precautionary loss function (PLF) as asymmetric loss functions.The SELF, PLF and DLF introduced by Legendre 19 , Norstrom 20 and DeGroot 21 , respectively.For a given posterior, the general expressions of the BEs and PRs are presented in Table 1.

Expressions for BEs and PRs using SELF
After simplification, the closed form expressions of BEs and PRs are given below: (24) Vol.:(0123456789) where v = 1 , v = 2 and v = 3 for the UP, JP and IP, respectively.Also, the BEs and PRs under other three loss functions can also be derived.For sake of shortness, we have not given these derived BEs and PRs but presented upon request to the corresponding author.

Elicitation of hyperparameters
Elicitation is a process used to enumerate a person's prior professional knowledge about some unidentified quantity of concern which can be used to improvement any values which we may have.In Bayesian analysis, specification and elicitation of hyperparameters of prior density is a common difficulty.For different statistical models, different procedures for specification of opinions to elicit hyperparameters of prior distribution have been established.
Aslam 22 suggested different methods which are depend upon the prior predictive distribution (PPD).In his study, one method based on prior predictive probabilities (PPPs) for elicitation of hyperparameters is used.The rule of evaluation is to link PPD with professional's evaluation of this distribution and to select hyperparameters which make evaluation agree narrowly with a part of family.So, subsequent the rules of probability the professional would be consistent in elicitation of the probabilities.A few conflicts may arise which are not significant.
2 can be applied to elicit the hyperparameters ξ 1 and ξ 2 , where p 0 (z) represent the elicited PPPs and p(z) denote the PPPs considered by hyperparameters ξ 1 and ξ 2 .For elicitation, the above equations are simplified numerically in Mathematica.A method depend upon PPPs is considered to elicit the hyperparameters of the IP In this study. ( Table 1.The BEs and PRs under loss functions.

Monte Carlo simulation study
From the Bayes estimators' expressions, it is clear that analytical assessments between BEs (using priors and loss functions) are not suitable.Therefore, the Monte Carlo simulation study is used to assess the presentation of BEs under various loss functions and priors.Moreover, the presentation of BEs has been checked under sample sizes and test termination time.We calculated the BEs and PRs of a 3-CMPD through a Monte Carlo simulation as: 1. From given 3-component mixture distribution, a sample consists of n p 1 , n p 2 and n 1 − p 1 − p 2 values out of n values is taken randomly from f 1 y , f 1 y and f 3 y , respectively.2. Select values which are larger than t as the censored values.The selection of t has been prepared in such a way that there is approximately 10% to 30% censoring rate in resultant data.Bayes estimates ωi and posterior risks ρ ωi of a parameter say ω are determine assuming censored values by solving ( 21)-(30).4. Repeat steps 1-3 for n = 30, 50, 100 , 1 , 2 , 3 , p 1 , p 2 = (0.4,0.3, 0.2, 0.5, 0.3) and t = 0.9, 0.6.
In case of choosing an appropriate prior, it is observed that IP materializes as an efficient prior because of lesser related PR as compared to NIP for estimating all five parameters under both symmetric and asymmetric loss functions (cf.Tables 2, 3, 4, 5, 6, 7, 8, 9).Also, it is noticed that JP (UP) emerges as a greater efficient because of smaller related PR as compared to UP (JP) for estimating component (proportion) parameters using both SELF and PLF (cf.Tables 2 and 6 vs Tables 4 and 8).Moreover, the UP is more efficient prior as compared to the JP under QLF and DLF due to smaller PR.On the other hand, the presentation of SELF is better than remaining three loss functions for estimating all parameters (cf.Tables 2 and 6).
It is also noticed that selection of an appropriate prior and loss function does not depend t.It is worth mentioning that our prior or loss function selection criterion is a posterior risk, i.e., we consider a loss function or prior the best if it yields minimum posterior risk as compared to others.

A real-life application
Here, the analysis of a lifetime data to explain the procedure for practical situations is presented.Gómez et al. 23 stated a lifetime data on exhaustion fracture of Kevlar 373/epoxy with respect to fix pressure at 90% pressure level till all had expired.Gómez et al. 23 showed that data x can be modeled with an exponential mixture model.However, the y = exp (−x) as a transformation of an exponentially distributed data (x) provides the power random variable and we can use the resulting data to describe the proposed Bayesian analysis.The lifetime data are divided into three groups of values with 26 values from 1st subpopulation, next 25 values from 2nd subpopulation, and the last 25 values from 3rd subpopulation.To use type-I censored samplings, we used the 3.4 as a censoring time and noted down the x 1 = x 11 , ..., x 1u 1 , x 2 = x 21 , ..., x 2u 2 and x 3 = x 31 , ..., x 3u 3 failed values from subpopulations I, II and III, respectively.The remaining values, which were greater than 3.4, have been taken censored values from each subpopulation.At the end of test, we have the following numbers of failed values, u 1 = 22 , u 2 = 22 and u 3 = 21 .The remaining n − u = 11 values were assumed censored values, whereas u = 65 were the uncensored values, such that u = u 1 + u 2 + u 3 .The data have been summarized as below:     10.
It is noticed that from the results, given in Table 10, are appropriate with the results given in simulation study section.The presentation of the BEs using IP is shown better than NIP as a result of smaller associated PRs for estimating all parameters under the different symmetric and asymmetric loss functions.Also, the BEs assuming JP (UP) is observed more suitable prior than UP (JP) based on smaller PRs for estimating component (proportion) parameters under SELF and PLF (SELF, QLF, PLF and DLF).In addition, it is revealed that the SELF is preferable to PLF, QLF and DLF due to minimum PRs for estimating all parameters.

3 .
Find the simulated Bayes estimates and posterior risks as ω =

Table 10 .
The BEs and PRs using the real life data.

Table 11 .
AIC and BIC for different mixture distributions.